Differentiation of Power Functions and Exponentials

Power Functions

A power function is when we have any variable x to the power of any real number represented here as n.

$y=x^{n}$

In the formula sheet we are given the equation:

$\frac{dy}{dx}=nx^{n-1}$

So basically all we do is multiply our function by the power term and then minus one the power term by one.

Power Function Differentiation Example

Say we have x to the power of negative fraction:

$y=x^{-\frac{5}{2}}$

We still follow the same procedure of multiplying the function by the power term and then minus it by one.

$\frac{dy}{dx}=-\frac{5}{2}\times x^{-\frac{5}{2}-1}$

$\frac{dy}{dx}=-\frac{5}{2} x^{-\frac{7}{2}}$

Polynomials

A polynomial is basically a combination of power functions. So while they may appear to be more tricky, if we isolate each term, they are just as easy as the example above.

Polynomial Differentiation example:

Given the equation below find the derivative:

$y=3x^{2}+\frac{3}{x}+4$

you first need to rewrite it so that it looks more straightforward:

$y=3x^{2}+3x^{-1}+4$

Then just follow the simple rule: bring the power down, put it in front, and take one away from the power.

Lets start with first term:

$3*x^{2}\newline 2\times 3x^{2-1}\newline -3x$

The second term:

$3x^{-1}\newline -1\times 3x^{-1-1}\newline -3x^{-2}$

The plus 4 at the end of equation is eliminated, as it has no x component.

To find $\frac{dy}{dx}$ we simply add these terms together

$\frac{dy}{dx}=6x-3x^{-2}+0\newline =6x-3x^{-2}$

Exponential Functions

The exponential functions that we focus on in maths methods is $e^{x}$

This is Euler’s number to the power of a variable x and has a remarkable differentiation property

Exponential Differentiation

In the formula sheet we are provided with the very useful formula:

$\frac{d}{dx}(e^{ax})=ae^{ax}$

This is one of the easiest rules to follow as all you do is multiply the exponential component by the coefficient of x

Exponential Differentiation Example

Let’s try to find the derivative for this function

$y=e^{3x}\newline \frac{dy}{dx}=3 \times e^{3x} \newline =3e^{3x}$

If we follow the formula provided, it greatly simplifies the process

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